Quotient Rule 05 (non-scaffolded)

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Michael Proudman

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Michael Proudman 4 years, 10 months ago

Published this.

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Michael Proudman 4 years, 10 months ago

Created this as a copy of Quotient Rule 05.

Name	Status	Author	Last Modified
Quotient Rule 01	Ready to use	Michael Proudman	07/03/2022 08:07
Quotient Rule 02	Needs to be tested	Michael Proudman	15/03/2022 16:35
Quotient Rule 03	Needs to be tested	Michael Proudman	15/03/2022 16:43
Quotient Rule 04	Needs to be tested	Michael Proudman	15/03/2022 16:48
Quotient Rule 05	Needs to be tested	Michael Proudman	15/03/2022 16:48
Quotient Rule 06	Needs to be tested	Michael Proudman	15/03/2022 16:50
Quotient Rule 01 (non-scaffolded)	draft	Michael Proudman	07/12/2021 08:32
Quotient Rule 02 (non-scaffolded)	draft	Michael Proudman	07/12/2021 08:32
Quotient Rule 03 (non-scaffolded)	draft	Michael Proudman	07/12/2021 08:32
Quotient Rule 04 (non-scaffolded)	draft	Michael Proudman	07/12/2021 08:32
Quotient Rule 05 (non-scaffolded)	draft	Michael Proudman	07/12/2021 08:32
Quotient Rule 06 (non-scaffolded)	draft	Michael Proudman	07/12/2021 08:32

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Question statement

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

If $\large y=\frac{u}{v}$ then:

$\frac{dy}{dx}= \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}}$

Now it is time to get out your paper and pencil and try similar questions without the help. Carry out the same steps, lay it out the same way but you only need to input the final answer.

Give any introductory information the student needs.

			Name	Type	Generated Value

			aC1	integer	4
			aP	integer	5

			Name	Type	Generated Value

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A random number between and (inclusive) with step size

Description

Part a) Constant term for denominator

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

→ Used by:

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Parts

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1. Gap Mathematical expression
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Differentiate $\large y=\frac{e^{\var{aP}t}}{t+\var{aC1}}$

$\Large \frac{dy}{dx}=$ Unnamed gap

Advice

We are asked to differentiate:

$\large y=\frac{e^{\var{aP}t}}{t+\var{aC1}}$

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

$u$ is the numerator, the function "on top", $v$ is the denominator, the function "on the bottom".

$u=e^{\var{aP}t}$ $v=t+\var{aC1}$

Now, we need to use the approriate techniques to differentiate each of these, for $u$ we need the Exponential Rule and for $v$ we need the Power Rule:

if $\frac{d}{dx} [a^x] = a^x \ln{(a)}$

If you don't follow this, use your Table of Derivatives.

Applying this gives us:

$\large \frac{du}{dx}=\simplify{{aP}e^({aP}t)}$ and $\frac{dv}{dx}=1$

Make the appropriatesubstitutions into the formula:

$\large \frac{dy}{dx}= \frac{(t+\var{aC1}) \times (\simplify{{aP}e^({aP}t)}) - e^{\var{aP}t} }{(t+\var{aC1})^2}$

Finally, we need to use our basic algebra terms to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

$\large \frac{dy}{dx}= \frac{\simplify{{aP}e^({aP}t)}(t+\var{aC1}) - (e^{\var{aP}t}) }{(t+\var{aC1})^2}$

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Checkpoint description

Michael Proudman 4 years, 10 months ago

Michael Proudman 4 years, 10 months ago

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

If $\large y=\frac{u}{v}$ then:

$\frac{dy}{dx}= \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}}$

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Differentiate $\large y=\frac{e^{\var{aP}t}}{t+\var{aC1}}$

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Quotient Rule 05 (non-scaffolded)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Michael Proudman 4 years, 10 months ago

Michael Proudman 4 years, 10 months ago

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

If y=uv\large y=\frac{u}{v} then:

dydx=vdudx−udvdxv2 \frac{dy}{dx}= \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}}

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Differentiate y=e{aP}tt+{aC1} \large y=\frac{e^{\var{aP}t}}{t+\var{aC1}}

If $\large y=\frac{u}{v}$ then:

$\frac{dy}{dx}= \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}}$

Generated value: `integer`

Differentiate $\large y=\frac{e^{\var{aP}t}}{t+\var{aC1}}$